This quantity's origin is from the first circular moment \(\mathbb {E}[e^{iX}]\)  which is decomposed into \(\phi _1=\rho e^{i\mu }\)  in the standard coordinate system of \(\mathbb {S}^1\) ; \(\rho \)  is the mean resultant length, and \(\mu \)  is also known as the mean direction [1]}. Before calculating \(\mu \) , it is paramount to determine what coordinate system one is working with. The first moment, \(\phi _1\) , is not necessarily always invariant to choice of coordinate system and \(\mathrm {Arg}\)  is only defined to have support in the standard coordinates of \(\mathbb {S}^1\) . One will have to convert \(\phi _1\)  to be in standard coordinates before proceeding to calculate \(\mu \) . Under the standard coordinates, one particular property \(\mu \)  has is that \(\mathbb {E}[\sin (X-\mu )]=0\) . This fact will be important in the following section.
